Parallax

Definition

Parallax is the apparent shift of a nearby object against a more distant background when the observer moves. Hold a finger at arm's length and look at it with one eye then the other: it jumps against the far wall. The shift is larger for closer objects, and that is how astronomers measure distances.

If the observer moves by a baseline B perpendicular to the line of sight and the object shifts by an angle p, its distance is d = B / tan p, or simply d ≈ B / p with p in radians. Astronomy uses two baselines: the Earth's radius for objects in the solar system (diurnal parallax), and the Earth's orbit for the stars (annual parallax).

Diurnal parallax in the solar system

As the Earth turns, an observer is carried across a baseline of up to one Earth radius. The horizontal parallax π is the angle under which the Earth's equatorial radius would be seen from the body; it is largest when the body is on the horizon:

sin π = 6378.14 km / Δπ (planet) = 8.794″ / Δ        (Δ in AU)

For the Moon, almost one degree, parallax is huge: two observers far apart see it against different stars, and the topocentric correction is needed for any accurate prediction of rising times, eclipses or occultations.

Measuring the astronomical unit

For centuries the key unknown was the solar parallax, which gives the Earth-Sun distance in kilometres. In 1672 Cassini and Richer measured the parallax of Mars from Paris and Cayenne and found a value close to the truth. The transits of Venus of 1761 and 1769 mobilised expeditions around the world, including Cook's voyage to Tahiti, to time the transit from distant places; they gave a solar parallax between about 8.5″ and 8.9″.

Today the astronomical unit is defined as exactly 149,597,870,700 m, measured by radar and spacecraft tracking, and the solar parallax is 8.794143″.

Stellar parallax and the parsec

As the Earth goes around the Sun, nearby stars trace small ellipses against more distant ones over a year. The annual parallax p is the semi-major axis of that ellipse, the angle under which the Earth-Sun distance is seen from the star. A star with a parallax of one arc second is at one parsec:

d (parsecs) = 1 / p (arc seconds)1 parsec = 206,265 AU = 3.26 light-years

Stellar parallaxes are tiny, which is why they were the great failure of astronomy until 1838: their absence was even used against the heliocentric system. Friedrich Bessel measured the parallax of 61 Cygni that year, about 0.3″ (0.286″ by modern measurements, a distance of 11.4 light-years). The largest stellar parallax, Proxima Centauri's, is only 0.768″: 4.24 light-years.

From Hipparcos to Gaia

Gaia's parallaxes have turned the Milky Way into a three-dimensional map. They anchor the whole cosmic distance ladder: Cepheid variables calibrated by parallax measure the distances to galaxies, which calibrate supernovae, which measure the expansion of the universe.

Parallax in Meeus's algorithms

Meeus devotes a chapter to the correction for parallax: from the geocentric right ascension, declination and distance, it computes the topocentric coordinates using the observer's latitude, height above sea level and the local hour angle. In this site's engine, the planets' horizontal coordinates go through this parallax correction, then through the correction for atmospheric refraction, before they are drawn in the horizon views.

Explore it on AstroAlgos

• Topocentric coordinates — correcting for parallax
• Semidiameter — apparent size and distance
• Night sky — the Moon at your location
• Geocentric coordinates